1. Any Mersenne number $M_p,~p ≡ 3 \pmod 4$ where $p$ is a prime, can be represented as: $$(8px+2p+1)(8py+1) = M_p,~0 ≤ x ≤ \frac{M_p - 1 - 2p}{8p},~ 0 ≤ y ≤ \frac{\frac{M_p}{2p + 1} - 1}{8p}$$
2. Any Mersenne number $M_p,~p ≡ 1 \pmod 4$ where $p$ is a prime, can be represented as: $$(8px+6p+1)(8py+1) = M_p,~0 ≤ x ≤ \frac{M_p - 1 - 6p}{8p},~ 0 ≤ y ≤ \frac{\frac{M_p}{6p + 1} - 1}{8p}$$
Could anyone prove or reject it?
Yes. Let's work through 1. first. Notice that $p \mid 2^{p-1} -1$ by Fermat's little theorem, so we can write $$2^{p-1} -1 = pm$$ for some $m$. Looking at that equation modulo 4, we see that $-1 \equiv -m \pmod 4$, i.e. $m=4k+1$ for some $k\geq0$.
Now pick $x=k$, $y=0$. We immediately get $$(8px+2p+1)(8py+1) = 2p(4x+1)+1 = 2pm+1 = 2^p -2 +1 = 2^p -1 = M_p.$$ It's pretty straightforward to check that the inequalities hold for $x$ and $y$.
For the second claim, you can construct a similar factorization with $y=0$, I'll leave the proof to you.